November  2017, 37(11): 5883-5912. doi: 10.3934/dcds.2017256

Visco-Energetic solutions to one-dimensional rate-independent problems

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy

Received  October 2016 Revised  June 2017 Published  July 2017

Visco-Energetic solutions of rate-independent systems (recently introduced in [17]) are obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation is reinforced by a viscous correction $δ$, typically a quadratic perturbation of the dissipation distance. Like Energetic and Balanced Viscosity solutions, they provide a variational characterization of rate-independent evolutions, with an accurate description of their jump behaviour.

In the present paper we study Visco-Energetic solutions in the scalar-valued case and we obtain a full characterization for a broad class of energy functionals. In particular, we prove that they exhibit a sort of intermediate behaviour between Energetic and Balanced Viscosity solutions, which can be finely tuned according to the choice of the viscous correction $δ$.

Citation: Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. Google Scholar

[2]

G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799. doi: 10.1142/S0218202502002331. Google Scholar

[3]

M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 13 (2006), 151-167. Google Scholar

[4]

I. S. Gál, On the fundamental theorems of the calculus, Trans. Amer. Math. Soc., 86 (1957), 309-320. doi: 10.1090/S0002-9947-1957-0093562-7. Google Scholar

[5]

D. Knees and A. Schröder, Computation aspect of quasi-static crack propagation, DCDS-S, 6 (2013), 63-99. doi: 10.3934/dcdss.2013.6.63. Google Scholar

[6]

K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67. doi: 10.1007/BF02416522. Google Scholar

[7]

D. Leguillon, Strength or toughness? A criterion for crack onset at a notch, European J. of Mechanics A/Solids, 21 (2002), 61-72. doi: 10.1016/S0997-7538(01)01184-6. Google Scholar

[8]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99. doi: 10.1007/s00526-004-0267-8. Google Scholar

[9]

A. Mielke, Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDEs and Applications, Lect. Notes Math, Springer, 2028 (2011), 87-170. doi: 10.1007/978-3-642-21861-3_3. Google Scholar

[10]

A. MielkeR. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems A, 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585. Google Scholar

[11]

Alexander MielkeRiccarda Rossi and Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80. doi: 10.1051/cocv/2010054. Google Scholar

[12]

Alexander MielkeRiccarda Rossi and Giuseppe Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc., 18 (2016), 2107-2165. doi: 10.4171/JEMS/639. Google Scholar

[13]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. Google Scholar

[14]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. Google Scholar

[15]

A. MielkeF. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194. Google Scholar

[16]

L. Minotti, Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016.Google Scholar

[17]

L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, arXiv: 1606.03359, (2016), 1-60.Google Scholar

[18]

M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925. doi: 10.1142/S0218202508003236. Google Scholar

[19]

R. RossiA. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 97-169. Google Scholar

[20]

R. Rossi and G. Savaré, A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 167-191. Google Scholar

[21]

T. RoubíčekC. C. Panagiotopoulos and V. Mantic, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840. doi: 10.1002/zamm.201200239. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. Google Scholar

[2]

G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799. doi: 10.1142/S0218202502002331. Google Scholar

[3]

M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 13 (2006), 151-167. Google Scholar

[4]

I. S. Gál, On the fundamental theorems of the calculus, Trans. Amer. Math. Soc., 86 (1957), 309-320. doi: 10.1090/S0002-9947-1957-0093562-7. Google Scholar

[5]

D. Knees and A. Schröder, Computation aspect of quasi-static crack propagation, DCDS-S, 6 (2013), 63-99. doi: 10.3934/dcdss.2013.6.63. Google Scholar

[6]

K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67. doi: 10.1007/BF02416522. Google Scholar

[7]

D. Leguillon, Strength or toughness? A criterion for crack onset at a notch, European J. of Mechanics A/Solids, 21 (2002), 61-72. doi: 10.1016/S0997-7538(01)01184-6. Google Scholar

[8]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99. doi: 10.1007/s00526-004-0267-8. Google Scholar

[9]

A. Mielke, Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDEs and Applications, Lect. Notes Math, Springer, 2028 (2011), 87-170. doi: 10.1007/978-3-642-21861-3_3. Google Scholar

[10]

A. MielkeR. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems A, 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585. Google Scholar

[11]

Alexander MielkeRiccarda Rossi and Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80. doi: 10.1051/cocv/2010054. Google Scholar

[12]

Alexander MielkeRiccarda Rossi and Giuseppe Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc., 18 (2016), 2107-2165. doi: 10.4171/JEMS/639. Google Scholar

[13]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7. Google Scholar

[14]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. Google Scholar

[15]

A. MielkeF. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194. Google Scholar

[16]

L. Minotti, Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016.Google Scholar

[17]

L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, arXiv: 1606.03359, (2016), 1-60.Google Scholar

[18]

M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925. doi: 10.1142/S0218202508003236. Google Scholar

[19]

R. RossiA. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 97-169. Google Scholar

[20]

R. Rossi and G. Savaré, A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 167-191. Google Scholar

[21]

T. RoubíčekC. C. Panagiotopoulos and V. Mantic, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840. doi: 10.1002/zamm.201200239. Google Scholar

Figure 1.  The double-well potential $W$ with its convex envelope in bold (left picture) and an energetic solution $u$ in the case of a strictly increasing load $\ell$ (right picture).
Figure 2.  BV solution for a double-well energy $W$ with an increasing load $\ell$. The blue line denotes the path described by the optimal transition $\vartheta$ solving (4).
Figure 3.  Visco-Energetic solutions for a double-well energy $W$ with an increasing load $\ell$. When $\mu>-\min W''$ (left picture) the solution jumps when it reach the maximum of $W'$ and the transition is the ''double chain'' obtained by solving the Incremental Minimization Scheme with frozen time $t$. When $\mu$ is small (right picture) the optimal transition $\vartheta$ makes a first jump connecting $\mathit{u}_{\tiny \mathsf L}(t)$ with $u_+$ according to the modified Maxwell rule (9): $\mathit{u}_{\tiny \mathsf L}(t)$ and $u_+$ corresponds to the intersection of $W'$ with the red line, whose slope is $-\mu$.
Figure 4.  The one-sided slopes and the stability region when $W$ is a double-well potential. For some suitable choices of $\delta$, $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ is intermediate between $W'_{\mathsf i\mathsf r}$ and $W'$.
Figure 5.  $\mathsf D$-Maxwell's rule for a double well potential: when $\delta=\frac{\mu}{2}|\cdot|^2$, the ''last point'' where $W'$ and $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ coincide is such that the total area between the graph $W'$ and the line whose slope is $-\mu$ is zero.
Figure 6.  Visco-Energetic solution of a double-well potential energy with an oscillating external loading and a quadratic viscous-correction $\delta(u,v)$, turned by a parameter $\mu>-\min W''$.
Figure 7.  Visco-Energetic solutions of a nonconvex energy and an increasing loading. The optimal transition is a combination of sliding and viscous parts.
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